论文标题
具有多$ 4 $ -CYCLES和BARNETTE的猜想的图形
Graphs with multi-$4$-cycles and the Barnette's conjecture
论文作者
论文摘要
令$ {\ cal h} $表示所有图表的家族都有多$ 4 $ -CYCLES，并假设$ g \ in {\ cal H} $中的$ g \。然后，$ g $是带有顶点两部分$ \ {v_α，v_β\} $的两部分图。我们证明，对于v_β$中的每个顶点$ v \，每$ 2 $颜色$v_α\ rightarrow \ rightarrow \ {1，2 \} $都存在$ 2 $ -Colouring $ -Orouring $v_β\ rightArrow \ rightarrow \ rightarrow \ rightArrow \ rightArrow \ rightArlow \ rightArlow \ rightArlow \ {1，2 \} $ in $ g $ in $ g $ in not nos go $ $ $ nes $ bon $ boy and（$ b） 现在让$ g $成为一个简单的平面三角剖分，带有顶点$ 3 $ - 分区$ \ {v_ {1}，v_ {2}，v_ {3} \} $。用$ b_ {i} $，$ i = 1、2、3 $表示，$ v_i $ a $ g $中的所有顶点的集合。假设$ g [b_ {1} \ cup b_ {3}] $（$ g [b_ {2} \ cup b_ {3}] $）是由集合$ b_ {1} \ cup b_ {1} \ cup b_ {3} $（$ b_ {3} $（$ b_ {2} \ cup b_ cup b_} $）的$ g $的子图。令$ g^{*} $为$ g $的双图，以下$ 3 $ -face-彩色：$ g^{*} $的face $ f $ of $ g^{*} $在$ i $时颜色为$ i $，并且仅当vertex $ v = f = f^{*} \ in v_ {i} $。我们证明，如果$ h = g [b_ {1} \ cup b _ {3}] \ cup g [b_ {2} \ cup b_ {3}] \ in {\ cal h} $，那么，对于在脸色$ 3 $上选择的任何边缘，那么对于$ 3 $和$ 6 $ in $ g^{避免这种边缘。此外，如果$ h $的每个组件都是$ 2 $连接的，则存在$ g^{*} $的汉密尔顿周期，以使每个面部有色$ 3 $都避免了这张脸的每一个边缘，或者在这张脸的最多两个边缘都避免了。
Let ${\cal H}$ denote the family of all graphs with multi-$4$-cycles and suppose that $G \in {\cal H}$. Then, $G$ is a bipartite graph with a vertex bipartition $\{V_α, V_β\}$. We prove that for every vertex $v \in V_β$ and for every $2$-colouring $V_α \rightarrow \{1, 2\}$ there exists a $2$-colouring $V_β \rightarrow \{1, 2\}$ such that every cycle in $G$ is not monochromatic and $b(v) = 1$ ($b(v) = 2$). Let now $G$ be a simple even plane triangulation with a vertex $3$-partition $\{V_{1}, V_{2}, V_{3}\}$. Denote by $B_{i}$, $i = 1, 2, 3$, the set of all vertices in $V_i$ of degree at least $6$ in $G$. Suppose that $G[B_{1}\cup B_{3}]$ ($G[B_{2}\cup B_{3}]$) is a subgraph of $G$ induced by the set $B_{1}\cup B_{3}$ ($B_{2}\cup B_{3}$, respectively). Let $G^{*}$ be the dual graph of $G$ with the following $3$-face-colouring: a face $f$ of $G^{*}$ is coloured with $i$ if and only if the vertex $v = f^{*} \in V_{i}$. We prove that if $H = G[B_{1}\cup B_{3}] \cup G[B_{2}\cup B_{3}] \in {\cal H}$, then, for any edge chosen on a face coloured $3$ and of size at least $6$ in $G^{*}$, there exists a Hamilton cycle of $G^{*}$ which avoids this edge. Moreover, if every component of $H$ is $2$-connected, then there exists a Hamilton cycle of $G^{*}$ such that for every face coloured $3$ it avoids every second edge of this face or it avoids at most two edges of this face.
